Objective
The student will be able to:
solve systems of equations using
substitution.
December 3, 2014
Solving Systems of Equations
You can solve a system of equations
using different methods. The idea is to
determine which method is easiest for
that particular problem.
 We previously worked on graphing.
There are two algebraic methods:
1. SUBSTITUTION
2. ELIMINATION

Special Systems of Linear Equations
Classification
Consistent and
Independent
Consistent and
Dependent
Inconsistent
Number of
Solutions
Exactly One
Infinitely Many
None
Description
Different Slopes
Same Slope,
Same y-intercept
They are the
same equation.
Same Slope,
Different
y-intercepts
Parallel lines
Graph
1) Solve the system using substitution
x+y=5
y=3+x
Step 1: Solve an equation
for one variable.
Step 2: Substitute
Step 3: Solve the equation.
The second equation is
already solved for y!
x+y=5
x + (3 + x) = 5
2x + 3 = 5
2x = 2
x=1
1) Solve the system using substitution
x+y=5
y=3+x
Step 4: Plug back in to find
the other variable.
Step 5: Check your
solution.
x+y=5
(1) + y = 5
y=4
(1, 4)
(1) + (4) = 5
(4) = 3 + (1)
The solution is (1, 4). What do you think the solution
would be if you graphed the two equations?
Which answer checks correctly?
3x – y = 4
x = 4y - 17
1.
2.
3.
4.
(2, 2)
(5, 3)
(3, 5)
(3, -5)
2) Solve the system using substitution
3y + x = 7
4x – 2y = 0
Step 1: Solve an equation
for one variable.
Step 2: Substitute
It is easiest to solve the
first equation for x.
3y + x = 7
-3y
-3y
x = -3y + 7
4x – 2y = 0
4(-3y + 7) – 2y = 0
2) Solve the system using substitution
3y + x = 7
4x – 2y = 0
Step 3: Solve the equation.
-12y + 28 – 2y = 0
-14y + 28 = 0
-14y = -28
y=2
Step 4: Plug back in to find
the other variable.
4x – 2y = 0
4x – 2(2) = 0
4x – 4 = 0
4x = 4
x=1
2) Solve the system using substitution
3y + x = 7
4x – 2y = 0
Step 5: Check your
solution.
(1, 2)
3(2) + (1) = 7
4(1) – 2(2) = 0
When is solving systems by substitution easier
to do than graphing?
When it is easy to isolate a variable in one of
the equations.
If you solved the first equation for x, what
would be substituted into the bottom equation.
2x + 4y = 4
3x + 2y = 22
1.
2.
3.
4.
-4y + 4
-2y + 2
-2x + 4
-2y+ 22
3) Solve the system using substitution
x=3–y
x+y=7
Step 1: Solve an equation
for one variable.
Step 2: Substitute
Step 3: Solve the equation.
There are NO SOLUTIONS.
The first equation is
already solved for x!
x+y=7
(3 – y) + y = 7
3=7
The variables were eliminated!!
This is a special case.
Does 3 = 7? FALSE!
3) Solve the system using substitution
2x + y = 4
4x + 2y = 8
Step 1: Solve an equation
for one variable.
Step 2: Substitute
Step 3: Solve the equation.
The first equation is
easiest to solved for y!
y = -2x + 4
4x + 2y = 8
4x + 2(-2x + 4) = 8
4x – 4x + 8 = 8
8=8
This is also a special case.
Does 8 = 8? TRUE!
When the result is TRUE, the answer is INFINITELY MANY SOLUTIONS.